High-temperature concomitant metal-insulator and spin-reorientation transitions in a compressed nodal-line ferrimagnet Mn3Si2Te6

Symmetry-protected band degeneracy, coupled with a magnetic order, is the key to realizing novel magnetoelectric phenomena in topological magnets. While the spin-polarized nodal states have been identified to introduce extremely-sensitive electronic responses to the magnetic states, their possible role in determining magnetic ground states has remained elusive. Here, taking external pressure as a control knob, we show that a metal-insulator transition, a spin-reorientation transition, and a structural modification occur concomitantly when the nodal-line state crosses the Fermi level in a ferrimagnetic semiconductor Mn3Si2Te6. These unique pressure-driven magnetic and electronic transitions, associated with the dome-shaped Tc variation up to nearly room temperature, originate from the interplay between the spin-orbit coupling of the nodal-line state and magnetic frustration of localized spins. Our findings highlight that the nodal-line states, isolated from other trivial states, can facilitate strongly tunable magnetic properties in topological magnets.


Response to Reviewer #1's comments
. The main observations are very similar to and different from that reported in Ref. 22, GPa, but the structural modification takes place at ∼12 GPa.Moreover, in a recent theoretical work (PHYSICAL REVIEW B 106, 045106 (2022)), the DFT calculations indicate that the critical pressure of the MIT depends on the value of LSDA + SOC +Ueff.Based on the LSDA + SOC +Ueff = 0.5 eV calculations, Zhang et al. found that the critical pressure for the MIT to be about 2.4 GPa, close to the experimental observation .
A1-1.We appreciate the helpful comments from the reviewer.As the reviewer pointed out, there is a clear deviation between our results and those of Wang et al [Phys. Rev. B, 106, 045106 (2022)].As described in detail below, we have conducted additional transport experiments on several different crystals, as well as infrared (IR) spectroscopy experiments.All the data consistently show that the metal-insulator transition (MIT) occurs at ~ 13-14 GPa, which is also in good agreement with the recent work conducted by a different group [arXiv: 2309[arXiv: .05945 (2023))].They claimed that MIT occurs above 10 GPa and certainly not at P ~ 1.5 -2.5 GPa as claimed by Wang et al. based on a single measurement on a single sample [Phys.Rev. B, 106, 045106 (2022)].
The critical pressure for the MIT at ~ 13-14 GPa as determined in our work, relies on measurements for six different Mn3Si2Te6 crystals from three different batches.We employed a standard criterion to determine the MIT critical pressure, where the system is inferred to be metallic when the slope of resistivity shows a positive temperature dependence (d/dT > 0).The crossover between insulating and metallic behaviors is clearly separated by the Mott-Ioffe-Regel (MIR) limit i.e. MIR = ħc/e 2 (depicted by the dashed line in Fig. R1a) where c is the caxis lattice constant (~14 Å) of Mn3Si2Te6.As presented in Fig. R1a, the pressure variation of room-temperature resistivity demonstrates excellent agreement between our different measurements with all data collapsing into a single curve.Additionally, we observed that the pressure-dependence resistivity estimated from the optical conductivity (see below) at low frequency, σ(ω=600 cm −1 ), is consistent with the transport results.Furthermore, the recent study by Huang et al [arXiv: 2309.05945 (2023)] claimed that MIT occurs above 10 GPa, which agrees well with our work.As shown in Fig. R1b, the pressure-dependence of normalized resistance ab(P)/ab (~ 1 GPa) by Huang et al [arXiv: 2309.05945 (2023)] follows a similar trend to our results albeit exhibiting a slightly different behavior at low pressures.In contrast, the data by Wang et al. show a rapid decrease at much lower pressures.
In order to further support our claim that MIT occurs at ~13-14 GPa, we measured the infrared (IR) reflectance spectra of Mn3Si2Te6 under pressure at room-temperature (Supplementary Fig S12).While a Drude-like mode appears in the low-frequency region of all spectra, recent findings indicate that this rise in the reflectance results from a phononic contribution occurring in the far-IR region below 600 cm −1 [arXiv:2311.14673(2023)].The reflectivity of Mn3Si2Te6 gradually increases with pressure, followed by a sharp rise above ~ 13 GPa, similar to the transport results.All reflectance spectra were then fitted using a Drude-Lorentz model employed in the RefFIT software [Rev. Sci. Instrum. 76, 083108 (2005)].The spectra can be well fitted with at least four Lorentz oscillators for P < 11 GPa, while an additional Drude mode was required to fit the spectra for P > 11 GPa.The corresponding optical conductivity σ(ω) was calculated via a standard Kramers-Kronig transformation from the fitted reflectance spectra (Fig. S12b).The optical conductivity σ(ω) increases gradually with pressure up to ~ 13 GPa, followed by a sudden enhancement at low energies indicating the contribution from free charge carriers.Similar behaviour was also observed in VO2 and attributed to the onset of pressureinduced MIT [Phys. Rev. Lett. 98, 196406 (2007)].The pressure-dependence of σ(ω = 600 cm −1 ), presented in Fig. S12c, exhibits a four-fold enhancement in the optical conductivity above ~13 GPa (arrow in Fig S12c ), consistent with the transport results.This optical conductivity enhancement near Pc provides spectroscopic evidence of the MIT in Mn3Si2Te6.Having established that the transport and IR spectroscopy results on our crystals, along with the recent findings by another independent group [arXiv: 2309[arXiv: .05945 (2023))], consistently reveal the MIT at much higher critical pressure than reported by Wang et al. [Phys. Rev. B, 106, 045106 (2022)], we now discuss the possible origins of this discrepancy.Like conventional semiconductors, a small amount of vacancies or impurities in Mn3Si2Te6 crystals, incorporated during the synthesis, introduce impurity (accepter) levels near the valence bands, providing charge carriers, as extensively discussed in Ref. 15 [Nature 599, 576-581 (2021)].Thus, the temperature-dependent and magnetic-field-dependent resistivity of Mn3Si2Te6 single crystals becomes highly sensitive to the impurity concentration within the crystal.The transport properties are mostly dictated by the transport activation gap  between the top of the valence bands and the Fermi level pinned by the impurity levels.The higher doping due to a larger amount of impurities indicates the smaller resistivity and magnetoconductivity.Unfortunately, a direct comparison of the resistivity values of our crystals with those of Wang et al's, is not feasible because they present the resistance of their crystals rather than the resistivity [Phys.Rev. B, 106, 045106 (2022)].Nevertheless, the magnetoconductivity at low temperatures, which is independent of the geometrical factors of the specimen, can be compared as shown in Fig. R2.
In FIG.R2, we presented the values of magnetoconductivity (MC) ratio at 10 K and 7 T for various Mn3Si2Te6 crystals reported in the literature [Phys.Rev. B, 103, L161105 (2021), Phys. Rev. B, 106, 045106 (2022), arXiv: 2309.05945 (2023)].Typically, the MC ratio at 10 K and 7 T, reflecting the MIT induced by the out-of-plane magnetic field is about 10 6 % or higher.However, the Mn3Si2Te6 crystal investigated by Wang et al. [Phys. Rev. B, 106, 045106 (2022)] exhibits the smallest MC ratio of ~10 4 % reported so far, which is at least two orders of magnitude lower than those reported previously, including from this work.The pressuredependent MC ratio of Mn3Si2Te6 crystal grown by Wang et al. also differs significantly as compared to the results of our work and the recent report [arXiv: 2309.05945(2023)], which agree well with each other.Consequently, the low MC ratio and its rapid reduction under pressure suggest that the Mn3Si2Te6 crystal grown by Wang et al. is highly doped due to a large impurity concentration.These observations raise concerns on the sample quality used in Wang et al's work, which may masks the intrinsic pressure-dependence of Mn3Si2Te6.Finally, we address a recent theoretical work by Zhang et al. [Phys. Rev. B, 107, 054430 (2023)] who identified the critical pressure for the MIT at 2.4 GPa, closely matching the experimental report, by setting the on-site Coulomb interaction of U = 0.5 eV.As emphasized in their report, the critical pressure of the MIT highly depends on the value of U, and the choise of U = 0.5 eV was specifically made to reproduce the existing experimental findings by Wang et al. [Phys. Rev. B, 106, 045106 (2022)].This means that the calculated critical pressure obtained with U = 0.5 eV should not be considered as an independent result supporting the findings of Wang et al.Moreover, the value of U = 0.5 eV reported in their work appears too small when compared to the U values appropriate for the Mn 3d states.In the case of other Mn-based chalcogenides, the typical U values are usually in the range of 3 -6 eV [Phys.Rev. B., 56, 7222 (1997), Phys. Stat. Sol. (b), 241, 1411(2004), Phys. Rev. Lett., 123 236401 (2019) and npj Quantum Mater. 5, 56 (2020)].Indeed, our first-principles calculations demonstrate that the critical pressure of MIT of ~13-14 GPa can be reproduced by considering U = 5-6 eV which falls in the typical range of the typical U values in other Mn chalcogenides.
In conclusion, we have unequivocally demonstrated that the MIT occurs at ~ 13-14 GPa, supported by on our transport measurements on several different crystals as well as IR spectroscopy experiments.This observation is consistent with the recent findings by an independent group [arXiv: 2309.05945 ( 2023)] confirming that the MIT occurs above 10 GPa, and certainly not between 1.5 -2.5 GPa as claimed by Wang et al. [Phys. Rev. B, 106, 045106 (2022)] based on a single measurement on a single sample.
In the revised Supplementary Information, we have provided additional results of IR measurements under pressure which supports the occurrence of MIT at P ~13-14 GPa.We also added discussion on the possible origin for the discrepancy between our work and Wang et al's work in the revised Supplementary Information.

Q1-2.
The main highlight of this manuscript could be the claim that the pressure-induced transitions occur concomitantly when the nodal-line state crosses the Fermi level in the ferrimagnetic semiconductor Mn3Si2Te6. However,from ref .22 and the DFT calculations, it is clear that the MIT could be decoupled with both SRT and structural modification.Moreover, the SRT can also be explained by the structural modification.A1-2.We thank the reviewer for this comment.In Section A1-1, we presented unequivocal evidence establishing the MIT at Pc ~13-14 GPa.As previously discussed in the original manuscript, the spin reorientation transition (SRT) and structural modification also occur at the same critical pressure indicating that all three transitions are intimately coupled.To further investigate the possible decoupling of the MIT and SRT, we carried out decompression experiments, during which we measured the transport properties upon releasing pressure from above Pc (Fig. S11).Although a significant hysteresis observed in the pressure dependent resistivity, the MIT behavior is reproduced during decompression.At P ~ 7.7 GPa during decompression, we observed the metallic behavior, accompanied by clear hysteresis and a square-shaped field dependence of the anomalous Hall conductivity, indicating the perpendicular magnetic anisotropy.However, after full decompression to P ~ 0.7 GPa, the insulating behavior is recovered.Simultaneously, the square-shaped anomalous Hall conductivity disappears and the magnetoresistivity becomes pronounced.These results suggest that spin direction is no longer along the c-axis and returns to the ab-plane upon full decompression.Notably, the initial structure of Mn3Si2Te6 is not entirely recoverable during decompression to ~ 1 GPa as seen by Raman spectroscopy.
These results firmly establish an intimate connection between SRT and MIT.Even with large strain disorder in the decompressed sample leading to significant variation of bond angles and distances between the neighboring Mn atoms significantly from the ideal values, the preferred spin orientation, either along the in-plane or the out-of-plane, is well defined and correlates with the electronic states, either gapped or gapless.Therefore, we can firmly conclude that indeed MIT and SRT are tied together, and the modification of crystal structure could be an additional consequence of the coupled MIT and SRT.
As discussed in the main text, the gap closing between the valence bands with nodal-line degeneracy and the conduction bands is essential for explaining the simultaneous occurrence of the MIT, SRT and structural transition.When the MIT transition is associated with such a gap closing, the charge transfer from the spin-split valence bands with spin-orbit coupling to the conduction bands can occurs, which triggers SRT and also induces moderate structural modification.In the case of the high doping as observed by Wang et al. [Phys. Rev. B, 106, 045106 (2022)], the MIT transition may occur by increasing the overlap between the impurity bands and the valence bands, without necessarily closing the electronic gap between the valence and conduction bands.Thus, to address the intrinsic pressure-dependent properties of Mn3Si2Te6, it is important to use the crystals with lower impurity levels as in our work [Nature 599, 576-581 (2021) ---------------------------------------------------------------------------------------------------------------Q2-1.In this manuscript, Susilo et.al. studied the transport, magnetic properties, and crystal structure of the ferrimagnetic semiconductor Mn3Si2Te6 under high pressures by using the diamond anvil cell.This work reveals that the pressure-induced metal-insulator transition is accompanied with the change of magnetic and crystal structures, undergoing a pressureinduced spin reorientation in the metallic state above the critical pressure Pc.The authors highlight the tuning of pressure on the nodal-line states and the physical properties of Mn3Si2Te6.In my opinion, this work is of high quality and deserves publication after addressing the following issues.
A2-1.We thank the reviewer for his/ her careful reading and for the positive assessment for our manuscript.

Q2-2.
The authors can easily determine the Tc at lower pressures by using the kink feature in resistivity.Can the authors use the same criteria to define Tc above Pc in the metallic state?A2-2.We appreciate the insightful comments from the reviewer.As the reviewer pointed out, the resistivity anomaly disappears above Pc in the metallic state, therefore a different criterion was used to determine Tc.As we have shown in the manuscript, once the system enters the metallic state above Pc, the field dependent Hall resistivity xy(H) shows a clear square-shaped behavior indicating the perpendicular magnetic anisotropy above Pc.Since   is dominated by the anomalous contribution, i.e.,   ≈    , the corresponding Hall conductivity,   (), can be scaled nicely with () as shown in Fig. S3.Assuming that the scaling factor   =    is nearly temperature-independent, the net magnetization () can therefore be represented by   () and thus Tc then can be determined from the temperature dependence of   .This approach enabled us to determine Tc above Pc leading to a complete magnetic phase diagram presented in Fig. 4c of the main text.
To make this part clearer to the reader, we have included relevant sentences in the revised manuscript.We also added the corresponding discussion in Supplementary Note 2. Q2-3.The authors stated that the evolution of Tc in this work is consistent with previous reports, e.g.Ref. 26, Phys. Rev. B 106, 045106 (2022).However, the behaviors of resistance are very different from that of Ref. 26.What is the cause for such differences?A2-3.We appreciate the helpful comments from the reviewer.As the reviewer pointed out, there is a clear deviation between our results and those of Wang et al [Phys. Rev. B, 106, 045106 (2022)].As described in detail below, we have conducted additional transport experiments on several different crystals, as well as infrared (IR) spectroscopy experiments.All the data consistently show that the metal-insulator transition (MIT) occurs at ~ 13-14 GPa, which is also in good agreement with the recent work conducted by a different group [arXiv: 2309.05945 ( 2023 The critical pressure for the MIT at ~ 13-14 GPa as determined in our work, relies on measurements for six different Mn3Si2Te6 crystals from three different batches.We employed a standard criterion to determine the MIT critical pressure, where the system is inferred to be metallic when the slope of resistivity shows a positive temperature dependence (d/dT > 0).The crossover between insulating and metallic resistivity curves is clearly separated by the Mott-Ioffe-Regel (MIR) limit i.e. MIR = ħc/e 2 (depicted by the dashed line in Fig. R1a) where c is the c-axis lattice constant of Mn3Si2Te6.As presented in Fig. R1a, the pressure variation of room-temperature resistivity demonstrates excellent agreement between our different measurements with all data collapsing into a single curve.Additionally, we observed that the pressure-dependence resistivity estimated from the optical conductivity (see below) at low frequency, σ(ω=600 cm −1 ), is consistent with the transport results.Furthermore, the recent data by Huang et al  In order to further support our claim that MIT occurs at ~13-14 GPa, we measured the infrared (IR) reflectance spectra of Mn3Si2Te6 under pressure at room-temperature (Supplementary Fig S12).While a Drude-like mode appears in the low-frequency region of all spectra, recent findings indicate that this rise in the reflectance results from a phononic contribution occurring in the far-IR region below 600 cm −1 [arXiv: 2311.14673 (2023)].The reflectivity of Mn3Si2Te6 gradually increases with pressure, followed by a sharp rise above ~ 13 GPa, similar to the transport results.All reflectance spectra were then fitted using a Drude-Lorentz model employed in the RefFIT software [Rev. Sci. Instrum. 76, 083108 (2005)].The spectra can be well fitted with at least four Lorentz oscillators for P < 11 GPa, while an additional Drude mode was required to fit the spectra for P > 11 GPa.The corresponding optical conductivity σ(ω) was calculated via a standard Kramers-Kronig transformation from the fitted reflectance spectra (Fig. S12b).The optical conductivity σ(ω) increases gradually with pressure up to ~ 13 GPa, followed by a sudden enhancement at low energies indicating the contribution from free charge carriers.Similar behaviour was also observed in VO2 and attributed to the onset of pressureinduced MIT [Phys. Rev. Lett. 98, 196406 (2007)].The pressure-dependence of σ(ω = 600 cm −1 ), presented in Fig. S12c, exhibits a four-fold enhancement in the optical conductivity above ~13 GPa (arrow in Fig S12c ), consistent with the transport results.This optical conductivity enhancement near Pc provides spectroscopic evidence of the MIT in Mn3Si2Te6 Having established that the transport and IR spectroscopy results on our crystals, along with the recent findings by another independent group [arXiv: 2309.05945(2023)], consistently reveal the MIT at much higher critical pressure than reported by Wang et al. [Phys. Rev. B, 106, 045106 (2022)], we now discuss the possible origins of this discrepancy.Like conventional semiconductors, a small amount of vacancies or impurities in Mn3Si2Te6 crystals, incorporated during the synthesis, introduce impurity (accepter) levels near the valence bands, providing charge carriers, as extensively discussed in Ref. 15 [Nature 599, 576-581 (2021)].Thus, the temperature-dependent and magnetic-field-dependent resistivity of Mn3Si2Te6 single crystals becomes highly sensitive to the impurity concentration within the crystal.The transport properties are mostly dictated by the transport activation gap  between the top of the valence bands and the Fermi level pinned by the impurity levels.The higher doping due to a larger amount of impurities indicates the smaller resistivity and magnetoconductivity.Unfortunately, a direct comparison of the resistivity values of our crystals with those of Wang et al's, is not feasible because they present the resistance of their crystals rather than the resistivity [Phys.Rev. B, 106, 045106 (2022)].Nevertheless, the magnetoconductivity at low temperatures, which is independent of the geometrical factors of the specimen, can be compared as shown in Fig. R2.
In FIG.R2, we presented the values of magnetoconductivity (MC) ratio at 10 K and 7 T for various Mn3Si2Te6 crystals reported in the literature [Phys. Rev. B, 103, L161105 (2021), Phys. Rev. B, 106, 045106 (2022), arXiv: 2309.05945 (2023)].Typically, the MC ratio at 10 K and 7 T, reflecting the MIT by the out-of-plane magnetic field is about 10 6 % or higher.However, the Mn3Si2Te6 crystal investigated by Wang et al. [Phys. Rev. B, 106, 045106 (2022)] exhibits the smallest MC ratio of ~10 4 % reported so far, which is at least two orders of magnitude lower than those reported previously, including our work.The pressure-dependent MC ratio of Mn3Si2Te6 crystal grown by Wang et al. also differs significantly as compared to the results of our work and the recent report [arXiv: 2309.05945(2023)], which agree well with each other.Consequently, the low MC ratio and its rapid reduction under pressure suggest that the Mn3Si2Te6 crystal grown by Wang et al. is highly doped due to a large impurity concentration.These observations raise concerns on the sample quality used in Wang et al's work, which may mask the intrinsic pressure dependence of Mn3Si2Te6.

Q2-4.
As described in the methods, the authors have used different pressure transmitting media for different high-pressure measurements.However, the results from different measurements seem to be quite consistent.Are there any influence of different medium on the physical properties?If so the authors may discuss briefly.A2-4.We thank the reviewer for this comment.As the reviewer mentioned, the use of different pressure media introduces varied and potentially severe stresses on the sample, particularly under non-hydrostatic conditions, where the sample experiences distinct stresses along different crystal directions.In some cases, this might affect the properties highly sensitive to stress gradient such as superconductivity.In general, the non-hydrostatic conditions are more likely to promote earlier or lower pressure-induced transitions.We agree with the reviewer that our results on Mn3Si2Te6 are consistent although different pressure media were used.This indicates that Mn3Si2Te6 is not highly sensitive to the non-hydrostatic condition.We note that such behavior is not unusual even in the case of layered materials where the interlayer spacing is important, as observed in SnS2 [RSC Adv., 12, 2454(2022)] and CrCl3 [Inorg. Chem., 61, 4852 (2022)].
Following the reviewer's suggestion, we included related sentences in the revised manuscript.

Q2-5.
The authors stated that a pressure-driven spin reorientation appears across the Pc, which can be extracted from the M(H) curves with obvious hysteresis loop for H//c at higher pressures above 12GPa.While for H//ab, the M(H) is only measured to 7.9GPa, as is shown in Fig. 2. Why the authors only increase the pressure to 7.9GPa?They should provide the M(H) curves at higher pressures to further confirm the pressure-induced spin reorientation with obvious reduction of magnetization for H//ab.A2-5.We appreciate this comment from the reviewer.The majority of the magnetization data for Mn3Si2Te6 were obtained using a modified turnbuckle diamond anvil cell (DAC) based on the design by Giriat et al. [Rev. Sci. Instrum., 81, 073905 (2010)].The dimensions of the sample space in the commercial Quantum Design SQUID MPMS magnetometer is approximately 10 mm long and 7 mm in diameter, which is compatible with the DAC for H//c configuration, where the magnetic field parallel to the direction of the applied load.To measure M(H) for H//ab, however, we had to use a much smaller DAC in length to rotate it by 90 degrees inside the sample space, which is more fragile at high pressures.This is the main reason why we only measured M(H) for H//ab up to ~8 GPa.
We made a new attempt to increase pressure close to Pc ~ 13-14 GPa, however the body of the pressure cell was not able to withstand the high load and broke instead.and employed for magnetic measurements at high pressures and low temperatures [Rev. Sci. Instrum., 82, 053906 (2011), Rev. Sci. Instrum., 86, 093901 (2015), High Press. Res., 37, 465 (2017)], all of them are typically capable of measuring magnetization with the magnetic field parallel to the direction of applied load.To the best of our knowledge, magnetic measurements with applied magnetic field perpendicular to the direction of load, as reported in this work, are quite rare.
Due to the technical limitation in measuring field-dependent magnetization for H//ab above P ~ 9 GPa, we turned to the anomalous Hall conductivity data (  () ) which shows a clear square-shaped field-dependence.This allowed us to show that the easy-axis of magnetization is no longer in the ab-plane, but instead along the c-axis above Pc.The square-shaped behavior in   () along the magnetic easy-axis is indeed common and have been reported in several topological magnets [Nature, 527, 212 (2015), Phys. Rev. Applied, 5, 064009 (2016), Nature Phys 14, 1125-1131(2018), Phys. Rev. Mater., 4, 044203 (2020), Nature 583, 533-536 (2020)].Therefore, we believe that a clear square-shaped hysteresis loop of   () with H // c , together with a clear jump in the field-dependent magnetization for H // c, provide sufficient evidence to justify that the Mn 2+ spin orientation is along the c-axis above Pc.Q2-6.As indicated by the high-pressure XRD patterns, a structural transformation appears at the Pc from trigonal structure to the monoclinic structure.However, the authors discuss the changes of electronic structures with the same crystal structure below and above the Pc.A2-6.We thank the reviewer for pointing out this issue, which was probably not clear in the original manuscript.As the reviewer mentioned the electronic structures of the low-pressure semiconducting trigonal phase should indeed differ from the high-pressure semimetal monoclinic phases.However, near the critical pressure Pc approaching from the lower pressure in the triclinic semiconducting phase, the nodal-line states in the valence bands at the  point becomes closer to the Fermi level.This proximity triggers the MIT, accompanied by the SRT and the structural modification as discussed in the main text.Thus, our discussion regarding the possible origin for the concomitant SRT and structural modification at Pc is based on the pressure-driven changes of the electronic structure in the low-pressure trigonal phase.
Nevertheless, we can find the signature of the nodal-line states even in the electronic structure of the semimetal monoclinic phase, just above Pc.As the MIT occurs at Pc ~ 13-14 GPa, we FIG.R4.Electronic band structures of Mn3Si2Te6 at 14 GPa for the high pressure crystal structure (C2/c space group) calculated with U = 5 eV.low temperature (T = 2 K) is in the order of 10 6 -10 7 Ω cm, which drops 7 -8 orders of magnitude, reaching   ()~10 -1 Ω cm under magnetic fields larger than 4 T [Phys.Rev. B, 103, L161105 (2021), Nature 599, 576-581 (2021), arXiv: 2309.05945 (2023)].In such a case, the MR ratio, as defined above, is nearly equal to -100% in the linear scale.While the   () monotonically and exponentially decreases with increasing pressure, the change in MR ratio in the linear scale becomes apparent only when   () is comparable to   (0) .The corresponding magnetoconductivity in the log scale, which can capture such a substantial resistivity change with magnetic fields, exhibits no distinctive anomaly in both our and Huang et al's results (FIG.R2b).Therefore, the seemingly sharp increase of the MR at ~ 6 GPa in the linear scale is a consequence of the exponential and monotonic pressure dependence of magnetoconductivity, which should not be interpreted as a signature of metal-insulator transition.
In our previous reply, we indeed showed that by choosing U = 5 eV in our DFT calculations yield the critical pressure for MIT to be Pc ~ 12 GPa, consistent with our experimental findings.In order to further support our claim on spin reorientation at Pc, we have calculated the total energy difference between spin orientations along [110]  While additional theoretical investigations are required for a comprehensive understanding of magnetic anisotropy and its pressure dependence, our calculations clearly show that a more appropriate choice of the on-site Coulomb energy U allows us to reproduce both experimentally-identified metal-insulator and spin orientation transitions at high pressures.
As discussed above, the reviewer's concerns regarding a few issues in the previous studies can be resolved by careful examination on the experimental data and additional calculations.
Lastly, we would like to emphasize again that, in addition to the aforementioned results, we have provided several experimental findings supporting our claims.In the previous response and revised manuscript, we shown that the MIT occurs at Pc ~14 GPa, based on our six measurements on three different crystals as well as IR spectroscopy experiments (Supplementary Notes 10 and 11), which is also further supported by the recent work by Huang et al. [arXiv: 2309.05945 (2023)] discussed above.The critical pressure at which the MIT occurs matches well with the SRT, observed from magnetization experiments.Moreover, in the previous revised manuscript, we have provided a critical finding from decompression experiments.Specifically, we observed that the square-shaped anomalous Hall conductivity, a signature of the perpendicular magnetic anisotropy, disappears as the insulating state of Mn3Si2Te6 is recovered on releasing pressure below 1 GPa, establishing an intimate connection between SRT and MIT.This observation implies that the preferred spin orientation, either along the in-plane or the out-of-plane, is well defined and correlates with the electronic states, either gapped or gapless (Supplementary Note 9).All these experimental evidences unequivocally support our claim and firmly establishes that MIT and SRT are coupled together and occur simultaneously at high pressures.
FIG. R1.(a) Pressure dependence of room temperature in-plane resistivity of Mn3Si2Te6 crystals studied in this work.For comparison, we included the resistivity value estimated from the optical conductivity at low frequency σ(ω = 600 cm −1 ) which is consistent with the transport data.Dashed line represents the estimated Mott-Ioffe-Regel limit.(b) Comparison of the pressure-dependent in-plane resistivity, normalized by its value at P ~ 1 GPa, for different Mn3Si2Te6 crystals.

Fig. S12 .
Fig. S12.Infrared spectroscopy studies of Mn3Si2Te 6 at high pressure.(a) Room-temperature reflectance spectra of Mn3Si2Te6 at various pressures.The data between 1600 cm −1 and 2700 cm −1 are excluded due to strong diamond absorption (indicated by dashed lines).(b) Real part of the complex optical conductivity (σ(ω)) at various pressures derived from the fitting to the reflectance spectra.(c) Pressure-dependent optical conductivity at ω = 600 cm −1 .A four-fold enhancement of the conductivity above ~13 GPa is indicated by the arrows.

Fig
Fig. S11.Physical properties of Mn3Si2Te 6 on decompression from high pressure above Pc.(a) Temperaturedependent ab-plane resistivity ρab(T) of Mn3Si2Te6 at various pressures during increasing and releasing pressure.Arrows indicate the resistive anomaly at Tc.(b) The magnetoresistance (MR) ratio (∆ρ/ρ(H)) and (c) Hall conductivity (σxy) at 15 K measured at various pressures.(d) Pressure-dependent resistivity of Mn3Si2Te6 at 300 K showing a significant hysteresis between compression and decompression.(e) Room temperature Raman spectra of Mn3Si2Te6 at various pressures.The disappearance of Raman modes on compression above Pc is retained during decompression to P ~ 1.4 GPa.In all Figures, label "D" represents data taken on decompression.
)].They claimed that MIT occurs above 10 GPa and certainly not at P ~ 1.5 -2.5 GPa as claimed by Wang et al. based on a single measurement on a single sample [Phys.Rev. B, 106, 045106 (2022)].
[arXiv: 2309.05945(2023)] claimed that MIT occurs above 10 GPa, which agrees well with our work.As shown in Fig. R1b, the pressure-dependence of normalized resistance ab(P)/ab (~ 1 GPa) by Huang et al [arXiv: 2309.05945(2023)] follows a similar trend to our results albeit exhibiting a slightly different behavior at low pressures.In contrast, the data by Wang et al.only show a rapid decrease at much lower pressures.
FIG. R1.(a) Pressure dependence of room temperature in-plane resistivity of Mn3Si2Te6 crystals studied in this work.For comparison, we included the resistivity value estimated from the optical conductivity at low frequency σ(ω = 600 cm −1 ) which is consistent with the transport data.Dashed line represents the estimated Mott-Ioffe-Regel limit.(b) Comparison of the pressure-dependent in-plane resistivity, normalized by its value at P ~ 1 GPa, for different Mn3Si2Te6 crystals.

Fig. S12 .
Fig. S12.Infrared spectroscopy studies of Mn3Si2Te 6 at high pressure.(a) Room-temperature reflectance spectra of Mn3Si2Te6 at various pressures.The data between 1600 cm −1 and 2700 cm −1 are excluded due to strong diamond absorption (indicated by dashed lines).(b) Real part of the complex optical conductivity (σ(ω)) at various pressures derived from the fitting to the reflectance spectra.(c) Pressure-dependent optical conductivity at ω = 600 cm −1 .A four-fold enhancement of the conductivity above ∼13 GPa is indicated by the arrows.
FIG. R3. (a-c) Additional field-dependent magnetization (M(H)) of Mn3Si2Te6 measured up to 8.5 GPa.Arrows indicate the saturation magnetic field.(d) Pressure-dependent of magnetocrystalline anisotropy energy K. (e) Pressure-dependent saturation magnetization Msat and the low-field c-axis magnetization Mc for H//c at 5 K.